Numerical Determination

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numerical determination

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  • Journal of Materials Processing Technology 216 (2015) 472483

    Contents lists available at ScienceDirect

    Journal of Materials Processing Technology

    jo ur nal home p ag e: www.elsev ier .com/ locate / jmatprotec

    Numerical determination of the forming limit cusheet m

    Abdolvah aninDorel Baa Mechanical E 3 Tehb CERTETA Rese -Napo

    a r t i c l

    Article history:Received 1 JulReceived in reAccepted 20 OAvailable onlin

    Keywords:GTN damage mForming limit FormabilityAnisotropy

    Need. Then isoe ABnt cod usility otherm

    are obtained using MarciniakKuczynski (MK) and modied maximum force criterion (MMFC) models.The results show that the forming limit curve predicted by the anisotropic GTN model is in better agree-ment with the experimental results especially in the biaxial tension region. This fact indicates that theGTN model is a useful tool in analyzing the formability of anisotropic sheet metals.

    2014 Elsevier B.V. All rights reserved.

    1. Introdu

    The formin industriea plot of theacterizing tpossible comand unsafe stand belowstrain comb

    Since th(1963), maexperimentexpenses intion, the thand numerito researchemination o

    CorresponE-mail add


    http://dx.doi.o0924-0136/ ction

    ing limit curve (FLC) is a very useful and common tools involved in sheet metal forming. This curve is actually

    major principal strain vs. minor principal strain char-he onset of sheet necking. Consequently, FLC divides the

    binations of the major and the minor strains into saferegions. More precisely, the strain combinations which

    the FLC are considered as safe (acceptable), while theinations located above the FLC are considered as unsafe.e introduction of the FLC by Keeler and Backhofenny attempts have been made to construct it usingal, theoretical and numerical methods. Because of thevolved by the experimental procedures of FLC construc-eoretical (Hill, 1952; Marciniak and Kuczynski, 1967)cal (Li et al., 2010) methods have been more attractivers. One of the suitable theoretical approaches for deter-

    f the FLC is the GursonTvergaardNeedleman (GTN)

    ding author. Tel.: +98 21 64543413/66419736.resses:, (A. Kami),.ir (B.M. Dariani), (A. Sadough Vanini), (D.S. Comsa), (D. Banabic).

    damage model (Gurson, 1977; Tvergaard, 1981, 1982; Tvergaardand Needleman, 1984). The original formulation of this model hasbeen proposed by Gurson (1977) by assuming that the degra-dation of the load carrying capacity and nally the fracture ofductile metals are caused by the evolution of voids. Gursonsmodel takes into account only the growth of pre-existing voids,without assuming any generative mechanisms. In order to over-come this limitation, Tvergaard (1981, 1982) and Tvergaard andNeedleman (1984) have proposed mathematical descriptions of thevoid nucleation and coalescence. The nal modied model is knownas GursonTvergaardNeedleman (GTN) damage model.

    As the metallic sheets are commonly produced by rolling, theyexhibit high levels of anisotropy. Due to this characteristic, it is veryimportant to include the anisotropy of the matrix material in theGTN model. There are few works that have dealt with this aspect.Liao et al. (1997) utilized a similar approach to that proposed byGurson (1977) to derive an approximate potential formulation forthe prediction of damage in the metallic sheets. The original fea-ture of their model consists in the fact that the equivalent stress isdescribed by Hills quadratic (Hill, 1948) and non-quadratic (Hill,1979) anisotropic expressions. Liao et al. (1997) used anisotropyparameters dened as the ratio of the transverse plastic strain rateto the through-thickness plastic strain rate under in-plane uni-axial loading along different directions. Wang et al. (2004) replaced

    rg/10.1016/j.jmatprotec.2014.10.0172014 Elsevier B.V. All rights reserved.etals using GTN damage model

    ed Kamia, Bijan Mollaei Dariania,, Ali Sadough Vnabicb

    ngineering Department, Amirkabir University of Technology, 424 Hafez Ave, 15875-441arch Centre, Technical University of Cluj-Napoca, Str. C. Daicoviciu nr. 15, 400020 Cluj

    e i n f o

    y 2014vised form 15 October 2014ctober 2014e 29 October 2014


    a b s t r a c t

    In this paper, the GursonTvergaardlimit curve of anisotropic sheet metalsHill48 quadratic yield criterion and ahas been developed and used inside thconstitutive model in the nite elemeAA6016-T4 sheet metal is constructesimulation of Nakjima tests. The quaexperimental forming limit curve. Furrves of anisotropic

    ia, Dan Sorin Comsab,

    ran, Iranca, Romania

    leman (GTN) damage model is used to determine the forming mechanical behavior of the matrix material is described usingtropic hardening rule. For this purpose, a VUMAT subroutineAQUS/Explicit nite element code. The implementation of thede is presented in detail. Finally, the forming limit curve of anng the developed VUMAT subroutine and running numericalf the numerical results is evaluated by comparison with anore, theoretical forming limit curves of the AA6016-T4 sheet

  • A. Kami et al. / Journal of Materials Processing Technology 216 (2015) 472483 473

    the directional parameters in the model proposed by Liao et al.(1997) with the average anisotropy parameter. Chen and Dong(2008) extended the GTN model to characterize the matrix materialthrough Hill quadratic (Hill, 1948) and BarlatLian 3-component(Barlat andand Dong (on Hills qu(Hill, 1948)

    Of coursto predict ttion. On thconstruct thoccurrence sheet and aanisotropic1981). Brunposed a necstrain localiis dened bBarlat and Lthe formingthe GTN mpredict theblank madeage model other sortset al., 2013;(Parsa et al(Uthaisangs

    In this psion of thelimit curvethe ABAQUmanual, 20fraction dismaterial padeterminedthe responsuniaxial ten

    2. FormulaAbaqus/Exp

    Abaqus/models by muses corotastrain as intime derivaformulationany concerassumes ththe plastic owith the fraverse direcThe followiquantities:

    ij comp

    arable into

    ij = (e)ij +

    ij componp hydros

    p = 3

    Hill48 equivalent stress:


    ijPijkk, (3)

    where Pijkl are components of a fourth-order tensor by means ofthe csheetstra


    othmatrequield s

    Y[ elass lawE

    + E andij de

    plasial (C






    rositantital costatelasti



    qs. (6






    (10)comet m

    p+ Lian, 1989) expressions of the equivalent stress. Chen2009) proposed extensions of the GTN potential basedadratic anisotropic expression of the equivalent stress.e, it is possible to solely use the GTN damage modelhe fracture of the ductile materials during deforma-e other hand, the GTN model could be also used toe forming limit curve. Brunet et al. (1996) studied theof necking in square cup deep drawing of a mild-steellso extracted the limit strains of the sheet using an

    GursonTvergaard criterion (Gurson, 1977; Tvergaard,et et al. (1998) and Brunet and Morestin (2001) pro-king criterion based on the load-instability and planezation assumptions in which the failure of the materialy GursonTvergaard damage model with Hill (1948) andian (1989) anisotropy models. He et al. (2011) predicted

    limit stress diagram of 5052 aluminum alloy based onodel. Abbasi et al. (2012a, 2012b) used GTN model to

    forming limit curve of an IF-steel and a tailor welded from IF-steel, respectively. Furthermore, the GTN dam-has been employed to predict the forming limits of

    of sheets like the AA5052/polyethylene/AA5052 (Liu Liu and Xue, 2013) and AA3105/Polypropylene/AA3105., 2013) sandwich sheets, dual-phase and multi-phaseuk et al., 2009; Ramazani et al., 2012) steels.aper, a GTN model based on Hills quadratic expres-

    equivalent stress is used to construct the forming. The model is implemented as a VUMAT routine inS/Explicit nite-element code (ABAQUS analysis users11). Furthermore, the plastic strain and void volumetributions near the fracture section are analyzed. Therameters involved in the constitutive relationships are

    by means of an identication procedure that combinese surface methodology (RSM) and the simulation of asile test.

    tion of the constitutive model and itslicit implementation

    Explicit allows the implementation of solid materialeans of the VUMAT routine. Because Abaqus/Explicit

    tional components of the Cauchy stress and logarithmicput/output when communicating with VUMAT, plaintives of such tensor quantities can be involved in the

    of the rate-type constitutive relationships, withoutn about their objectivity. The model presented belowat the Abaqus/Explicit corotational frame also reectsrthotropy of the sheet metal, being initially coincidentme dened by the rolling direction RD (axis 1), trans-tion TD (axis 2) and normal direction ND (axis 3).ng symbols will denote macroscopic strain and stress

    onents of the corotational logarithmic strain tensor sep-

    elastic (e)ij

    and plastic (p)ij

    terms, i.e.



    ents of the corotational Cauchy stress tensortatic pressure:


    which of the the con

    Pijk = Two

    dense (p)

    Y yilaw Y =


    ij = 1wherewhile




    f (f ) =

    is a poThe qumateristress to the e

    Thethe for



    or, if E







    Eq. that acthe she